1. 9/22/2013PHY 711 Fall 2013 -- Lecture 121 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 12: Continue reading Chapter 3 & 6 1.Hamiltonian formalism 2.Phase space 3.Liouville’s theorem

2. 9/22/2013PHY 711 Fall 2013 -- Lecture 122

3. 9/22/2013PHY 711 Fall 2013 -- Lecture 123 Hamiltonian formalism

4. 9/22/2013PHY 711 Fall 2013 -- Lecture 124 Hamiltonian formalism and time evolution:

5. 9/22/2013PHY 711 Fall 2013 -- Lecture 125 Poisson brackets: Hamiltonian formalism and time evolution:

6. 9/22/2013PHY 711 Fall 2013 -- Lecture 126 Poisson brackets -- continued: Liouville theorem

7. 9/22/2013PHY 711 Fall 2013 -- Lecture 127 Phase space

8. 9/22/2013PHY 711 Fall 2013 -- Lecture 128 x p Phase space diagram for one-dimensional motion due to constant force

9. 9/22/2013PHY 711 Fall 2013 -- Lecture 129 Phase space diagram for one-dimensional motion due to spring force

10. 9/22/2013PHY 711 Fall 2013 -- Lecture 1210 Liouville’s Theorem (1838) The density of representative points in phase space corresponding to the motion of a system of particles remains constant during the motion.

11. 9/22/2013PHY 711 Fall 2013 -- Lecture 1211 Liouville’s theorem x p (x,p) (x+  x,p) (x+  x,p+  p) (x,p+  p)

12. 9/22/2013PHY 711 Fall 2013 -- Lecture 1212 Liouville’s theorem -- continued x p (x,p) (x+  x,p) (x+  x,p+  p) (x,p+  p)

13. 9/22/2013PHY 711 Fall 2013 -- Lecture 1213 Liouville’s theorem -- continued x p (x,p) (x+  x,p) (x+  x,p+  p) (x,p+  p)

14. 9/22/2013PHY 711 Fall 2013 -- Lecture 1214 Importance of Liouville’s theorem to statistical mechanical analysis: In statistical mechanics, we need to evaluate the probability of various configurations of particles. The fact that the density of particles in phase space is constant in time, implies that each point in phase space is equally probable and that the time average of the evolution of a system can be determined by an average of the system over phase space volume.

15. 9/22/2013PHY 711 Fall 2013 -- Lecture 1215 Virial theorem (Clausius ~ 1860)